Simplicial spaces, lax algebras and the -Segal condition
Abstract
Dyckerhoff–Kapranov [5] and Gálvez-Carrillo–Kock–Tonks [7] independently introduced the notion of a -Segal space, that is, a simplicial space satisfying -dimensional analogues of the Segal conditions, as a unifying framework for understanding the numerous Hall algebra-like constructions appearing in algebraic geometry, representation theory and combinatorics. In particular, they showed that every -Segal object defines an algebra object in the -category of spans.
In this paper we show that this algebra structure is inherited from the initial simplicial object . Namely, we show that the standard -simplex carries a lax algebra structure. As a formal consequence the space of -simplices of a simplicial space is also a lax algebra. We further show that the -Segal conditions are equivalent to the associativity of this lax algebra.
1 Introduction
Simplicial objects satisfying the Segal conditions [26] are used throughout homotopy theory and higher category theory to encode coherently associative algebras. Recently, Dyckerhoff–Kapranov [5] and Gálvez-Carrillo–Kock–Tonks [7] independently introduced a generalisation of Rezk’s Segal conditions, the -Segal conditions. They showed that simplicial objects satisfying the -Segal conditions encode algebra objects in -categories of spans. In this paper we shall elucidate the exact role played by the -Segal condition in the construction of these algebra objects.
Specifically, we provide a novel construction of the algebra in the -category of spans associated to a -Segal object. We do so by exploiting the fact that the initial simplicial object in an -category having finite limits is
where is the nerve of the category of level-wise finite simplicial sets. The following diagram
![[Uncaptioned image]](/html/1710.02742/assets/x1.png){kind=small} |
(1.1) |
endows the standard -simplex with a product as an object of the -category of spans in . While the product fails to be associative, the diagram
![[Uncaptioned image]](/html/1710.02742/assets/x2.png) |
defines a non-invertible -morphism in the -category of bispans [10] in from to which witnesses its lax associativity. We show the following:
Theorem 1.
The standard -simplex is a lax algebra in the -category of bispans in with product .
From any simplicial object in an -category having finite limits one has a finite limit preserving functor given by right Kan extension,
The simplicial object is then the image of under the right Kan extension. As a formal consequence of Theorem 1 one obtains a lax algebra structure on as an object of the -category of bispans in . We show that the -Segal condition is equivalent to the associativity of this lax algebra.
Theorem 2.
The object of -simplices of a simplicial object is canonically a lax algebra in the -category of bispans in . Furthermore, satisfies the -Segal conditions if and only if is an algebra object.
To our knowledge there are two previous constructions of the above algebra associated to a -Segal object by different techniques and at differing levels of generality. The first is due to Dyckerhoff–Kapranov, who associate to each injectively fibrant -Segal object in a combinatorial simplicial model category a monad on the -category of bispans which makes an algebra when ([5] 11.1). Gálvez-Carrillo–Kock–Tonks showed the space of -simplices in a -Segal space carries an algebra structure using a novel equivalence between simplicial spaces and certain monoidal functors ([7] 7.4).
By first constructing the lax algebra for the initial simplicial object we are able to work at the highest level of generality possible, that of -Segal objects in an -category having finite limits. Furthermore, this paper is the first in a series of papers which build new examples of higher categorical bialgebras using -Segal spaces [24, 23]. The construction given here provides the groundwork for these later papers.
Outline.
The paper begins in Section 2 with the necessary background for defining the notion of a lax algebra object in a symmetric monoidal -category. As the informal description of a lax algebra given above explicitly makes use of non-invertible -morphisms it should come as no surprise that the definition makes use of ideas which are inherently -categorical. Specifically, one needs lax functors between -categories. In the particular model of -categories that we use in this paper lax functors are defined in terms of the unstraightening construction, the -categorical generalisation of the ordinary Grothendieck construction. A lax algebra object in a symmetric monoidal -category is then defined to be a symmetric monoidal lax functor out of the category which corepresents algebra objects. In Section 2.1 we review our chosen model of -categories, in Section 2.2 we define lax functors in terms of the unstraightening construction and in Section 2.3 we introduce the category .
As indicated in Theorems 1 and 2 we will be constructing lax algebra objects in symmetric monoidal -categories of bispans, which are the subjects of Section 3. One of the technical hurdles that one must surmount when working with lax functors of -categories is the difficulty of providing an explicit description of the unstraightening construction. As such, we devote a considerable part of this section to the unstraightening of the -category of bispans, rendering it into a form which is adequate for our purposes. We begin, in Section 3.1, with a brief discussion on the twisted arrow construction, a construction which appears throughout this work. Next we review Haugseng’s original definition of the -category of bispans in Section 3.2. Finally, Sections 3.3 and 3.4 contain technical material culminating in a description of the unstraightening of the -category of bispans.
While the previous two sections have been essentially preparatory, Section 4 presents the main results of the paper. We begin in Section 4.1 by proving Theorem 1, that is, by giving an explicit construction of a lax algebra structure on . The first statement in Theorem 2 is proven in Section 4.2 as a formal consequence of the construction in the previous section. Finally, in Section 4.3 we prove the second statement of Theorem 2 connecting the -Segal condition to the associativity of the lax algebras constructed in Section 4.2.
Acknowledgements.
We would like to thank Tobias Dyckerhoff for a number of insightful conversations on the subject of -Segal spaces. We also thank Rune Haugseng for answering some of our questions regarding the unstraightening construction and the -category of bispans. Finally, we are indebted to Joachim Kock for his thorough feedback on an earlier draft of this paper.
Notational conventions and simplicial preliminaries.
Throughout this paper we make extensive use of the theory of -categories as developed by Joyal [13, 14] and Lurie [19]. In particular, by an -category we shall always mean a quasi-category, a simplicial set having fillers for all inner horns.
Ordinary, simplicially-enriched or model categories will always be denoted by either Greek letters (e.g. ) or in ordinary font (e.g. ). -categories will either be blackboard Greek letters (e.g. ) or have the first character in calligraphic font (e.g. ).
For ordinary categories and , is the category of functors and natural transformations and is the groupoid of functors and natural isomorphisms. Analogously, for -categories and , is the -category of functors and is the largest Kan complex inside . In particular, the (-)category of -fold simplicial objects in an ordinary category or -category are
where is the category of non-empty linearly ordered finite sets and the -category is the nerve of .
Let denote the simplicially-enriched category of quasi-categories with mapping spaces given by . The -category of -categories, , is the coherent nerve of ([19] 3.0.0.1). The full subcategory of of Kan complexes is denoted , and its coherent nerve, , is the -category of spaces. The inclusion admits a right adjoint ([19] 1.2.5.3, 5.2.4.5) denoted
We denote by css the category of bisimplicial sets carrying the Rezk model structure [26]. This is a simplicial model category whose fibrant-cofibrant objects are the complete Segal spaces. The coherent nerve of the full subcategory of complete Segal spaces, denoted , is canonically equivalent to ([15] 4.11).
The category of finite sets is denoted by and every object is isomorphic to one of the form
Similarly, is the category of pointed finite sets. Objects of are denoted , with the basepoint and the complement of the basepoint. The -categories and are, respectively, the nerves of and .
We adopt the topologists convention for the category of non-empty, linearly ordered finite sets in that we label its objects by
The active and inert morphisms form a factorization system on . The former are those morphisms which preserve the bottom and top elements while the latter are the inclusions of subintervals. Active morphisms will be denoted by arrows of the form () while inert morphisms by arrows of the form (). Their corresponding wide subcategories are denoted, respectively, by and . Every morphism in can be uniquely factored as an active followed by an inert morphism. The -categories and are, respectively, the nerves of and .
Remark 1.1.
The active-inert factorisation of morphisms in is a particular example of the general notion of generic-free factorisations in the theory of monads as developed by Weber [28] and Berger-Mellies-Weber [4]. Following Lurie [18] and Haugseng [11], we adopt the former terminology as we feel that it is more descriptive.
The category is a full subcategory of , the category of (possibly empty) linearly ordered finite sets, the objects of which are
The category ([8] 8)222Note that our category is the opposite of the one defined in [8] has the same objects as and morphisms given by spans of the form
| (1.2) |
The category of -fold nabla objects in a category is
There is a bijective on objects and full functor which restricts to isomorphisms and ([8] 8.2). Restriction along induces a fully faithful functor
| (1.3) |
The category has a canonical monoidal structure
having unit . The functor restricts to a monoidal equivalence if we endow with the monoidal structure
| (1.4) |
having unit .
Finally, throughout this paper algebra objects are assumed to be unital.
2 Lax algebras in symmetric monoidal -categories
In this section we cover the necessary background for defining the notion of a lax algebra object in a symmetric monoidal -category. We begin in Section 2.1 with a review our chosen model of -categories. In Section 2.2 we define lax functors in terms of the unstraightening construction. Finally, in Section 2.3 we introduce the category which corepresents algebras.
2.1 Review of symmetric monoidal -categories
The model of -categories that we use in this work is the one originally introduced by Lurie in [20].
Definition 2.1.
An -category is a simplicial -category such that
-
1.
is a Segal object, that is, for each the functor is an equivalence;
-
2.
The -category is a space, that is, ; and,
-
3.
The Segal space is complete.
The -category of -categories is a full subcategory . In other words, functors between -categories are simply natural transformations.
Remark 2.2.
A widely used model of -categories in the literature are -fold complete Segal spaces as introduced by Barwick [1]. Barwick’s model is recovered from the one used in this work by presenting as .
Let denote the automorphism sending to .
Definition 2.3.
For an -category its opposite -category is the composite
There is a universal way to extract from a Segal object in a new Segal object whose -category of -simplices is a space: let denote the full subcategory of spanned by Segal objects and let be the full subcategory of the former spanned by those objects satisfying Condition above. Then the inclusion admits a right adjoint ([10] 2.13)
Explicitly, given , we have and for each a pullback square
| (2.1) |
There are a number of ways in the literature to define symmetric monoidal -categories. We follow Lurie ([18] 2.0.0.7) in choosing the one generalising Segal’s notion of a special -space [27].
Definition 2.4.
A symmetric monoidal -category is a functor
such that
-
1.
For each , the simplicial -category is an -category.
-
2.
For each and , the map is an equivalence.
The -category of symmetric monoidal -categories is a full subcategory
In other words, a symmetric monoidal functor between symmetric monoidal -categories is simply a natural transformation, that is, a morphism in .
2.2 Symmetric monoidal lax functors via the unstraightening construction
Recall that a lax functor between ordinary -categories differs from a functor in that it no longer respects identity arrows and composition of -morphisms [16]. Instead, for each object of and each pair of composable morphisms and , one has (not necessarily invertible) -morphisms and in witnessing the lax preservation of unitality and composition. Furthermore, these -morphisms must satisfy associativity and unitality coherence equations.
The notion of a lax functor between -categories requires the use of the theory of cocartesian fibrations of -categories and the unstraightening construction as developed by Lurie ([19] 2.4). Recall that the unstraightening construction defines an equivalence between functors and cocartesian fibrations over ([19] 3.2.0.1),
Under this equivalence a natural transformation is sent to a morphism
such that sends cocartesian morphisms in to cocartesian morphisms in . The unstraightening construction is natural in in the sense that from a composite
one has a pullback diagram ([9] A.31)
Taking one can unstraighten an -category, and functors become morphisms of cocartesian fibrations over which preserve cocartesian morphisms. Lax functors will still be morphisms of fibrations but will preserve fewer cocartesian morphisms.
Definition 2.5.
A lax functor between -categories is a morphism
such that sends cocartesian lifts of morphisms in , the subcategory of inert morphisms, to cocartesian morphisms.
Remark 2.6.
We are not certain as to the exact history of this approach to defining lax functors between -categories. We first learned it from Dyckerhoff–Kapranov ([5] 9.2.8) and Lurie’s definition of a morphism of -operads ([19] 2.1.2.7) is qualitatively similar. We believe an analogous statement must be known for lax functors between -categories but do not know any references.
Remark 2.7.
Since preserves inert morphisms, the naturality of the unstraightening construction implies that from a lax functor one gets a lax functor .
It is straightforward to extend this notion to define symmetric monoidal lax functors, that is, functors between symmetric monoidal -categories which preserve the symmetric monoidal structure but only laxly preserve composition, as follows.
Definition 2.8.
A symmetric monoidal lax functor between symmetric monoidal -categories is a morphism
such that sends cocartesian lifts of morphisms in to cocartesian morphisms.
2.3 Corepresenting algebra objects
To specify an algebra object in a symmetric monoidal -category one must provide not just an associative and unital binary operation, but a coherent choice of higher associativity and unitality data. To package together all of this data we will make use of a category originally introduced by Pirashvili [25].
Denote by the category having as objects finite sets. A morphism in is a function along with a choice of linear ordering of the (possibly empty) preimages for each . The composition of a pair of composable morphisms
is the composition of the underlying functions, with linear ordering on given by
where the sum denotes the monoidal structure on , the category of finite linear orders, introduced in Eq. 1.4. The disjoint union endows with a symmetric monoidal structure.
The category corepresents algebra objects in the sense that, for a symmetric monoidal category , symmetric monoidal functors are the same as algebra objects in . This is because is the category of operators [22] for the -operad of associative algebras, that is, and all morphisms in are, up to precomposition with an isomorphism, disjoint unions of these.
Remark 2.9.
One only needs a monoidal structure on a category to define algebra objects in it. The simpler category corepresents algebra objects in monoidal categories and so one can equivalently define an algebra object in a symmetric monoidal category to be a monoidal functor from to . However, one needs at least a braiding on a monoidal category to define bialgebra objects in it, and in all the examples which concern us the braiding is in fact symmetric. As this paper lays the foundations for our work on higher categorical bialgebras [24, 23] it is therefore crucial that we make use of rather than .
To corepresent algebra objects in a symmetric monoidal -category it suffices to present in the model of symmetric monoidal -categories that we use in this paper. Observe that for an -category and a fully faithful functor, both the right and left Kan extensions,
are fully faithful should they exist.
Definition 2.10.
Let be an -category and be a fully faithful functor. We call functors in the image of cartesian and those in the image of cocartesian. We denote these full subcategories, respectively, by and , with the functor to be understood implicitly from the context.
Remark 2.11.
We follow Haugseng’s terminology [10] as our definition of a (co)cartesian functor is a generalisation of the one given there. In Section 2.2 we discussed cocartesian morphisms and cocartesian fibrations. These are distinct notions from the one being introduced now, but as these terms are used in different contexts there is little fear of confusion.
For each set , denote by the poset of subsets of ordered by inclusion. These assemble into a functor by declaring the image of a pointed map to be
For each there is a full subcategory consisting of the singleton subsets. The -categories and are, respectively, the nerves of and .
Example 2.12.
A functor is a diagram,
Such a diagram is cartesian if it presents as the product of and and is terminal. Similarly, a cartesian functor encodes a coherent choice of products for a collection of objects of labelled by the elements of .
Recall that the symmetric monoidal structure on an -category having finite coproducts is given by the functor
While the disjoint union is not the coproduct in , it is the case that given morphisms there is a unique morphism making the following diagram commute
Define a functor to be cocartesian if for each , the composite
is cocartesian. One has by the above that the functor
sending to the set of cocartesian functors is a symmetric monoidal -category.
We can now define the algebraic structures which are the focus of the remainder of this paper.
Definition 2.13.
Let be a symmetric monoidal -category.
-
•
An algebra object in is a symmetric monoidal functor .
-
•
A lax algebra object in is a symmetric monoidal lax functor .
Remark 2.14.
One can readily dualise the preceding discussion to define coalgebra objects in symmetric monoidal -categories. Namely, a (lax) coalgebra object in is a symmetric monoidal (lax) functor from to .
It is worth taking a moment to informally discuss the exact nature of a lax algebra object , as it is slightly more subtle than one might initially expect. For each string of composable morphisms in ,
one has -morphisms
which are compatible with disjoint union and the composition of morphisms in . In particular, the witness to the lax associativity of the product on is a diagram
3 The -category of bispans
The construction which associates the -category to an -category having finite limits can be iterated to form -categories for each . Informally, a -morphism between spans is a ‘span of spans’, that is a diagram of the form
while a -morphism is a ‘span of spans of spans’, and so on. The cartesian product in endows these -categories with a symmetric monoidal structure. The rigorous construction of these symmetric monoidal -categories has been carried out by Haugseng [10].
Section 3.1 is a brief discussion on the twisted arrow construction, a construction which appears throughout this work. In Section 3.2 we review Haugseng’s construction in the case that concerns us, namely the construction of the symmetric monoidal -category of bispans in . In Section 3.3 we prove that is semistrict, a technical condition which simplifies the description of lax functors. Finally, in Section 3.4 we determine explicitly the unstraightening of for an ordinary category having finite limits.
3.1 The twisted arrow category
The twisted arrow category ([21] IX.6.3), , of an ordinary category will be a recurring character in this work. It will first appear in the definition of the -category of bispans in Section 3.2 and its subsequent reappearances will be tied to various constructions involving bispans.
For an ordinary category , let be the category having as objects arrows in , and morphisms from to
Remembering only the source and target of an object of defines a forgetful functor . Furthermore, the twisted arrow categories assemble into a functor .
Remark 3.1.
A number of constructions in later sections involve writing explicit functors of the form for a category having finite limits. It turns out that such functors can be equivalently described as normal oplax functors , where is a bicategory which we shall describe shortly. For our purposes this latter description will often be more convenient.
Given a category having finite limits one can define a bicategory [3] having the same objects as , -morphisms given by spans, and -morphisms given by diagrams
Horizontal composition is given by pullbacks, chosen once and for all. The identity -morphism for an object is the span .
Remark 3.2.
Note that this bicategory is similar to, but distinct from, the -category of bispans in that we shall introduce in Section 3.2. They have different -morphisms and, unlike the -category of bispans in , has no -morphisms for .
A normal oplax functor consists of, for each object an object , for each morphism in a span
and for each pair of composable morphisms a diagram
For each object it must be that , for each morphism it must be that , and, suppressing associator isomorphisms, for each string of composable morphisms ,
| (3.1) |
as morphisms .
Theorem 3.3 ([6] 3.4.1).
For any category and any category having finite limits there is a natural bijection
where denotes the set of normal oplax functors.
The isomorphism is given as follows. A normal oplax functor associates to each diagram
a diagram
The corresponding functor is then
Finally, an immediate corollary of Theorem 3.3 is the following:
Corollary 3.4.
Let denote the category of categories having finite limits and finite limit preserving functors between them. Then for any diagram , the functors
are naturally isomorphic.
Proof.
Since bicategories and normal oplax functors between them assemble into a category, the set-valued functor sends colimits in the first variable to limits in sets. Let be a diagram in . By Theorem 3.3 one has for each having finite limits the following string of natural bijections
∎
3.2 Definition of
Haugseng’s construction of the symmetric monoidal -category is an iteration of Barwick’s construction of the -category [2]. One first defines a symmetric monoidal double -category which fails to be in . One then remedies this problem by defining .
The morphisms in the -category of bispans are given by diagrams in of the following form. Let be the twisted arrow construction of . Explicitly, this is the opposite of the poset of non-empty intervals in . Similarly, the category is opposite of the poset of non-empty rectangles in . These assemble into a functor
There is a full subcategory consisting of intervals with and hence a full subcategory . The -categories and are, respectively, the nerves of and .
Example 3.5.
A functor is a diagram,
Such a diagram is cartesian if it is the right Kan extension of its restriction to , i.e., if the middle square is a pullback in . In general, a functor is pyramid of spans on objects. It being cartesian says higher tiers of this pyramid consist of a coherent choice of pullbacks of the spans along the bottom two tiers.
Presenting as we can now define the symmetric monoidal -category of bispans. For an -category having finite limits, consider the functor
which is well-defined by Proposition 3.8 of [10]. Taking the coherent nerve defines a functor .
Proposition 3.6.
Let be an -category with finite limits. Then the functor
is a symmetric monoidal -category. We call this the symmetric monoidal -category of bispans in .
Remark 3.7.
Our description of the symmetric monoidal structure arising from the cartesian product on differs from Haugseng. He instead presents the symmetric monoidal structure by giving a sequence of -categories delooping the -category of bispans ([10] 9.1).
Proof.
By Corollary 6.5 of [10], for each fixed , the simplicial -category is a -category. Consider the following commutative diagram in ,
where denotes the fully faithful functor including into . Since preserves limits it suffices to show that the top map is an equivalence.
By Definition 2.10, the category is the image of under , the right adjoint of . Since is fully faithful it is an equivalence onto its image with inverse . It follows that the top map in the above diagram is an equivalence, as desired. ∎
3.3 is semistrict
Describing symmetric monoidal lax functors for general symmetric monoidal -categories can be quite complicated due to the use of the unstraightening construction. For the following class of symmetric monoidal -categories it simplifies greatly.
Recall that a functor is equivalent to a functor of simplicially enriched categories, where is the left adjoint of the coherent nerve ([19] 1.1.5).
Definition 3.8.
For an ordinary category , a functor is called semistrict if there is a functor of ordinary categories , for the ordinary category of quasi-categories and functors between them, such that the following commutes
where is the counit of the adjunction . In particular, a symmetric monoidal -category is semistrict if it is semistrict as a functor .
The aim of this section is to show that is semistrict. To that end, we must first compute the pullback of -categories in diagram 2.1, or equivalently, the homotopy pullback of
| (3.2) |
in css, the category of bisimplicial sets endowed with the Rezk model structure [26].
The computation is rendered trivial by the following lemma.
Lemma 3.9.
For each , the morphism
is a fibration in css.
Proof.
It suffices to show that is a Reedy fibration since css is a left Bousfield localisation of the Reedy model structure on the category of bisimplicial sets, and both the source and the target of are fibrant in css ([12] 3.3.16).
For the purposes of this proof we will use the shorthand . Consider the following commutative diagram in ,
| (3.3) |
The morphism is a Reedy fibration if and only if the induced map from to the pullback is a Kan fibration of simplicial sets ([26] 2.4).
Let be the pushout of simplicial sets
The pullback of diagram 3.3 is since sends colimits to limits. To prove that is a Kan fibration it suffices to show that
is a Kan fibration as the target of is a union of connected components of the target of .
By left Kan extension along the Yoneda embedding one can extend to a functor . Let to be the edgewise subdivison functor sending to ([2] 2.5). Since and both functors preserve colimits one has that . From this we conclude that preserves products and monomorphisms.
It follows, therefore, that the target of is . The morphism is induced by the functor , which is itself induced by the inclusion and so is a monomorphism. Hence, by Lemma 3.1.3.6 of [19], is a Kan fibration. ∎
Next, denote by the sub-poset of on those morphisms of the form . We say a functor is vertically constant if morphisms of the form for are sent to equivalences.
Definition 3.10.
Let be an -category with finite limits and . Define to be the simplicial set having -simplices the cartesian, vertically constant functors .
Proposition 3.11.
Let be an -category with finite limits. Then for each in , the simplicial set is a quasi-category. Furthermore, the symmetric monoidal -category of bispans is given by the functor
and so is semistrict.
Proof.
Recall that for a complete Segal space each vertex of determines a sequence of composable morphisms in the homotopy category of . Then is the full sub simplicial set of on those vertices having each invertible in the homotopy category ([20] 1.1.11).
In particular, is the full sub simplicial set of generated by those functors sending morphisms of the form to equivalences. Since all of the objects in diagram 3.2 are fibrant, Lemma 3.9 implies that the ordinary pullback in css is a complete Segal space presenting the homotopy pullback ([19] A.2.4.4). We can therefore conclude that is the full sub simplicial set of generated by those functors sending morphisms of the form for to equivalences.
The canonical equivalence is presented by a Quillen equivalence ([15] 4.11)
By definition , which is a quasi-category since preserves fibrations. ∎
Corollary 3.12.
For any -category having finite limits its symmetric monoidal -category of bispans is equivalent to its opposite.
Proof.
By Proposition 3.11, the opposite of the symmetric monoidal -category of bispans is
where the quasi-category has -simplices the cartesian and vertically constant functors . Since is canonically isomorphic to , one has an equivalence and hence . ∎
3.4 Unstraightening
The aim of this section is to give a description, sufficient for our purposes, of the unstraightening of the symmetric monoidal -category of bispans for an ordinary category having finite limits. In general, determining the quasi-category unstraightening a functor can be quite difficult. This can, however, be done in the special case when for an ordinary category and is semistrict ([19] 3.2.5.2). In this case, the set of -simplices is the set of pairs
| (3.4) |
where is the maximal element of . The family of functors must be such that for each the following diagram commutes
By Proposition 3.11, is semistrict for any -category having finite limits, in particular when as we shall assume for the remainder of this section333In fact, we only ever make use of the particular case of . Performing a similar analysis as we present in this section for a general -category would involve explicitly determining certain colimits in . This is both considerably more difficult and unnecessary for our purposes.. We can therefore apply the above to compute its unstraightening.
Remark 3.13.
As a slight abuse of notation we will not distinguish between an element and its opposite functor
Furthermore, for we write the composite morphisms as
Let be the wide subcategory of on the injective maps and let denote the full subcategory of on those objects such that . For each , denote by the composite
where is the forgetful map from to .
Lemma 3.14.
The -simplices of the unstraightening of are the pairs
such that each component is cartesian and vertically constant.
Proof.
By Eq. 3.4 and Proposition 3.11, a -simplex in the unstraightening of is an element along with, for each , a cartesian and vertically constant functor
such that for the following diagram commutes
| (3.5) |
where the functor along the top is induced by and the functor along the left is induced by . Note that for every string of composable morphisms in one can build an analogous commutative -cube based on the commutativity of diagram 3.5 and the functoriality of .
From this family we shall construct a natural transformation . For each one has a morphism , and hence the diagram 3.5 with replaced by commutes. We define to be the composite functor along the diagonal of this diagram, which is cartesian and vertically constant by construction. Given a morphism there is a triple of composable morphisms
The naturality of is a consequence of the commutative -cube constructed from this triple of composable morphisms.
Conversely, given a natural transformation having cartesian and vertically constant components, define . The commutativity of diagram 3.5 follows directly from the naturality of . ∎
By the universal property of colimits in , a natural transformation is equivalently a functor . Observe that the diagonal inclusion of into the full subcategory of on objects , where , is final. Therefore is isomorphic to the product of the colimits over and the diagram
It follows that
where is the composite
| (3.6) |
It follows from Corollary 3.4 that functors are equivalently functors .
Our first task in this section is to compute . We shall then determine conditions under which a functor restricts to cartesian and vertically constant functors .
Computing the colimit of the diagram .
Recall that the Grothendieck construction of a functor is the category having as objects
and morphisms
Let , for , be the category which is the Grothendieck construction of the functor
Explicitly, is the poset having object set
and ordering defined by declaring if and only if and .
Example 3.15.
For the constant map on , the poset is .
Example 3.16.
For the unique active morphism , the poset is
Lemma 3.17.
The poset is the colimit of the diagram .
Proof.
As the category is obtained via the Grothendieck construction it is the colimit of the diagram
Observe that, since , one can obtain this diagram by precomposing the diagram in Eq. 3.6 with the functor
The functor is final, as for any with , the category is the full subcategory of on those objects
which is non-empty and connected. ∎
The cartesian and vertical constancy conditions.
Having determined that we shall now define conditions under which a functor induces cartesian and vertically constant functors for each .
Definition 3.18.
Define to be the sub-poset of on those morphisms and to be the full sub-poset of on those intervals satisfying and .
Example 3.19.
For the constant map on , one has that and .
Example 3.20.
For the unique active morphism , the poset is
and the poset is
Definition 3.21.
Let be a category with finite limits and . Then we say a functor is:
-
1.
cartesian if it is the right Kan extension of its restriction to .
-
2.
vertically constant if morphisms of the form for are sent to isomorphisms.
Remark 3.22.
When is the constant map on this definition reduces to the notions already introduced for functors .
Since the categories for various are obtained by the Grothendieck construction they are compatible in the following sense. For a natural transformation one has a functor
and for a morphism there is a functor
Lemma 3.23.
Let be an category having finite limits, and a cartesian functor. Then for a natural transformation and a morphism , the composite functors
are cartesian.
The proof of this Lemma is somewhat technical and the details are unnecessary for the remainder of the text. We therefore separate its proof into Section 3.4.1.
Proposition 3.24.
Let be a category with finite limits, and let be the set
Then the sets assemble into a sub simplicial set of the unstraightening of .
Remark 3.25.
In general, is a proper sub simplicial set of the unstraightening. This will nonetheless suffice for our purposes as we will make use of this explicit form to map into the unstraightening.
Proof.
To begin, we observe that by Lemma 3.23, if is cartesian then so is for any , where
Furthermore, since maps to , the functor is vertically constant whenever is. Therefore, the sets assemble into a simplicial set.
By Lemma 3.17 the set of -simplices of the unstraightening of consists of pairs
such that the induced functors are cartesian and vertically constant. To show that is a sub simplicial set it therefore suffices to show that if is cartesian and vertically constant than so are the functors . For the remainder of this proof we shall fix a cartesian, vertically constant functor .
Observe that is , where is the constant functor on , and that one has a diagram
where is the natural transformation having components . Letting denote the composite functor
one has that is
It follows from Lemma 3.23 that is cartesian. The vertically constancy of follows from noting that sends to and sends to . ∎
3.4.1 Proof of Lemma 3.23
Throughout this subsection we shall fix , a natural transformation , a morphism and a cartesian functor . We shall also make use of some notational shorthands. Set , , , , , , and .
We first show that is cartesian. Since is cartesian, for each , the object is the limit of the diagram
where is the full subcategory of containing as well as those objects such that for some . On the other hand, the right Kan extension of the restriction of to evaluated at is the limit of the diagram
To show that is cartesian it therefore suffices to show that for each , the functor is initial ([21] IX.3). That is, we must show that for each , the poset
is non-empty and connected.
Letting we define the following
There are a now two cases to consider. First, if then is the maximal element of satisfying . Therefore and so is non-empty and connected as it has terminal object . Second, if then from the inequalities
one concludes that , , and is the largest subinterval of sent to under . The poset is non-empty as it contains . Letting denote the maximal subinterval of such that , one has that for any there is at least one such that . As the poset contains the connected sub-poset
and each element maps into this sub-poset it follows that is connected.
Next, we will show that is cartesian. The argument is quite similar to the above, and we shall recycle certain notation. It suffices to show that for each and , the poset
is non-empty and connected, where is full subcategory of containing and everything above the image of under . Set and
There are again two cases. The first is when , then is the maximal element of satisfying , proving as above that is non-empty and connected. Next, if then and is the largest subinterval of sent to . The same analysis as above, mutatis mutundi, shows that is non-empty and connected.
4 Simplicial objects define lax algebras in
In this section we present the main results of this paper. We first, in Section 4.1, explicitly construct a symmetric monoidal lax functor
endowing the standard -simplex with the structure of a lax algebra. Then, in Section 4.2, we show how the object of -simplices of a simplicial object in an -category having finite limits inherits the same structure from the universal property of . Finally, we show in Section 4.3 that satisfies the -Segal condition if and only if it inherits an algebra structure from .
4.1 The lax algebra structure on
Our first step in the construction of the lax algebra coming from a general simplicial object will be to carry out the construction for the initial simplicial object . According to Definitions 2.8 and 2.13, this amounts to constructing a morphism of fibrations
preserving cocartesian lifts of morphisms of the form . Furthermore, the image of the object in the fibre over must be .
Before diving into the detailed construction of , let us first give an informal description. On objects, the lax functor is simply
Recall that a morphism in is a function along with a linear ordering of for each . The lax functor sends the morphism to the morphism given by the diagram
The morphism sends the -simplex associated to the element to the long edge of the standard simplex . Note that one can label the edges along the spine of by the elements of using the linear ordering. The morphism sends the -simplex associated to to the appropriate edge along the spine of .
Example 4.1.
For the morphism , the morphism in is given by the diagram in Eq. 1.1.
Example 4.2.
Consider the morphism . Then is given by the diagram
![[Uncaptioned image]](/html/1710.02742/assets/x3.png){kind=small} |
The lax structure on the functor is given by associating to each pair of composable morphisms in a -morphism in of the form
Example 4.3.
Consider the pair of composable morphisms . The lax structure on is given by the diagram
![[Uncaptioned image]](/html/1710.02742/assets/x4.png) |
Outline of the construction.
Recall from Section 2.3 that is semistrict by construction, and so it is straightforward to determine from Eq. 3.4 that its unstraightening is given by
From such data we will build a cartesian and vertically constant functor
By Proposition 3.24 this defines a -simplex in the unstraightening of . To better understand the approach we take to the construction of it will be instructive to describe a special case.
Consider where is constant on and is the unique active morphism. Then the functor is a pair of composable morphisms
Since is cartesian it is determined by its restriction to , the diagram described in Example 3.20. The restriction of to is
Observe that all of the data in this diagram can be obtained from the following diagram
| (4.1) |
which is a functor . Specifically, the restriction of to is obtained from by restricting along a functor .
In general, observe that for each one has a natural transformation having components . The naturality of the Grothendieck construction implies that this natural transformation induces a functor
Our construction of for general proceeds by first defining a functor
generalising the one in Eq. 4.1. The functor is then defined so that the following diagram is a right Kan extension
| (4.2) |
The functor is manifestly cartesian, and is vertically constant as every morphism in is sent to the identity in by .
Construction of the functor .
By Theorem 3.3 it suffices to define a normal oplax functor
Recall from Eq. 1.2 that is the category of spans of the form in . The category of levelwise finite nabla sets, denoted is the category of functors . From a morphism in one can define the following diagram of levelwise finite nabla sets
where and are constant nabla sets and is the nabla set represented by . The first and second map arise, respectively, from the following morphisms in :
We can then apply the functor of Eq. 1.3 to obtain a span in .
Now, define on objects as
To define on a morphism in , one applies the above construction to the morphism in , yielding a span in
From a pair of composable morphisms in , one has a commutative diagram of nabla sets
where and arise, respectively, from the following morphisms in :
Applying the functor of Eq. 1.3 yields a diagram in .
For a pair of composable morphisms in we define the corresponding component of the oplax structure on as follows. Applying the construction of the preceding paragraph to the image under of the pair of composable morphisms one has, in particular, a commutative square
| (4.3) |
Then the component is the universal morphism
induced by the diagram in Eq. 4.3.
Lemma 4.4.
For each , the above data defines a normal oplax functor and hence a functor
Proof.
The unitality conditions are straightforward to verify. Given a triple of composable morphisms in , along similar lines as above one has a diagram in
Then the associativity condition,
holds since both sides of the equation are the universal morphism for the colimit of the bottom two rows of the above diagram. ∎
The morphism is a symmetric monoidal lax functor.
The diagram in Eq. 4.2 defines, for each a functor
We define , for , to be
where is the sub simplicial set of the unstraightening of from Proposition 3.24.
Lemma 4.5.
The assignment defines a morphism of simplicial sets
Proof.
Fix and a morphism . We must show that , that is, that the following diagram commutes,
where is the composite
Letting denote the full subcategory of containing as well as those such that for some , one has that the following diagram commutes
It therefore suffices to show that for each ,
as normal oplax functors . This follows from the fact that is cocartesian. ∎
We have therefore constructed a morphism of fibrations
Furthermore, the image of the object in the fibre over is . To show that the endows with a lax algebra structure it remains only to show that defines a symmetric monoidal lax functor.
Proposition 4.6.
The morphism of simplicial sets is a symmetric monoidal lax functor
Hence, endows the standard -simplex with the structure of a lax algebra..
Proof.
We must show that for every with inert the functor
defines an equivalence in the quasicategory ([19] 3.2.5.2).
For inert, the poset is of the form
with the image of indicated in red.
Therefore for each the functor is the right Kan extension of a diagram in of the form
As the equivalences in are exactly those cartesian functors which send all morphisms of the form to equivalences ([10] 6.2) the claimed result follows. ∎
4.2 The lax algebra structure inherited from
Let be a simplicial object in an -category having finite limits. We can now show how inherits a lax algebra structure in from the one endowed upon standard -simplex in Proposition 4.6.
We make use of the following result due to Li-Bland.
Theorem 4.8 ([17] 4.1).
The construction which assigns to an -category having finite limits its symmetric monoidal -category of bispans defines a functor
where is the -category of -categories having finite limits and finite limit preserving functors between them.
Recall that any simplicial object defines a finite limit preserving functor by right Kan extension,
By Theorem 4.8 this induces a symmetric monoidal functor
Then as an immediate corollary of Proposition 4.6 one has the following.
Theorem 4.9.
Let be an -category with finite limits and let be a simplicial object. Then the composite
is a symmetric monoidal lax functor endowing with the structure of a lax algebra.
Remark 4.10.
Following Remark 4.7 we obtain a symmetric monoidal lax functor endowing with the structure of a lax coalgebra.
4.3 Associativity and the -Segal condition
Having equipped the object of -simplices of a simplicial object with a lax algebra structure in in Theorem 4.9, we now demonstrate that the -Segal condition is exactly the right condition that enforces the associativity of this structure.
Definition 4.11.
[5] 2.3.2 A simplicial object is a -Segal object if and only if for every and every , the image of the squares
| (4.4) |
under are pullbacks in and for each and , the image of the square
| (4.5) |
under is a pullback in . In particular, a -Segal space is a -Segal object in , the -category of spaces.
Remark 4.12.
Remark 4.13.
Theorem 4.14.
Let be an -category with finite limits. Then a simplicial object is a -Segal object if and only if the symmetric monoidal lax functor of Theorem 4.9 endows with the structure of a algebra.
Proof.
We must show that is a -Segal object if and only if is an equivalence in for every .
Every active morphism can be decomposed as
where is the unique active morphism and . Every inert morphism is of the form . Since morphisms in can be uniquely factored as an active morphism followed by an inert morphism, one can decompose the morphism as
One can therefore write the poset schematically as
![[Uncaptioned image]](/html/1710.02742/assets/x5.png){kind=small} |
Denoting by the restriction of to the -th summand of , it follows that for each , the functor is the right Kan extension of a diagram in of the form
![[Uncaptioned image]](/html/1710.02742/assets/x6.png) |
Therefore is an equivalence if and only if is an equivalence for each .
To conclude our proof of Theorem 4.14 we must prove two lemmas.
Lemma 4.15.
Let be a simplicial object in an -category having finite limits. If, for every with being the unique active morphism is an equivalence in , then satisfies the -Segal condition.
Proof.
Consider, for each and , the following pairs of composable morphisms in :
| (4.6) |
The morphism maps to when and otherwise, while maps to when and otherwise. The linear orders on the fibres are induced by the ordering on . Observe that
are, respectively, the pushouts of the left and right square of Eq. 4.4. Furthermore, , and the corresponding components of the lax structure on agree with the ones arising from the squares in Eq. 4.4.
Next, consider for each and the pair of composable morphisms in :
| (4.7) |
where is the evident map which skips the element in . Observe that
is the pushout of Eq. 4.5. We also have that , and the corresponding components of the lax structure on agree with the ones arising from the square in Eq. 4.5.
Finally, consider , where is constant on and is the unique active map. Then the functor is a pair of composable morphisms
and is the diagram
The functor is an equivalence in precisely when is an equivalence.
For the second of the two lemmas which complete Theorem 4.14 we must make use of an equivalent formulation of the -Segal condition due to Gálvez-Carrillo–Kock–Tonks ([7] 3). They show that is -Segal if and only if the image under of every pushout square in of the form
is a pullback in .
Lemma 4.16.
Let be a -Segal object. Then is an equivalence in for every with being the unique active morphism.
Proof.
It suffices to show that for every sequence of composable morphisms in
the image of
under is an equivalence. Since every morphism in is a disjoint union of morphisms having target the singleton set, and the functors and are symmetric monoidal, it suffices to consider the case when . The statement for general follows by iterating the special case of .
First, in the trivial case of , then also and
For , write and denote by
where is the inverse on objects of the functor in Eq. 1.3. Recall that can be thought of as a standard simplex having the edges along its spine labelled by the elements of according to the linear ordering defined by . The pushout is the simplicial set obtained by gluing each simplex by its long edge to the corresponding edge on the spine of . Therefore is the iterated pushout
Next, set to be the inductively defined pushouts in
| (4.8) |
where has image the smallest two elements of , and has image the ’th and ’th elements of where . Then since one has that . Furthermore, the morphism factors as the composite
| (4.9) |
where each morphism arises from the pushout squares in Eq. 4.8.
It follows from the Gálvez-Carrillo–Kock–Tonks form of the -Segal condition that each morphism the sequence in Eq. 4.9 is sent to an equivalence under , proving the claimed result. ∎
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